Abstract
In this paper we demonstrate that the well-known Hamilton-Jacobi method can be used in study of the rheo-linear (i.e. time dependent) harmonic oscillator with a single degree of freedom. It will be shown that the quadratic conservation laws (exact invariants) together with corresponding auxiliary equations follow immediately from the complete integral of Hamilton-Jacobi partial differential equation by application of the Jacobi theorem. Attention is also paid to the study of linear conservation laws and to the motion of rheo-linear dynamical systems.
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